EE359 Discussion Session 9 OFDM, Spread Spectrum ... · Narrowband interference rejection ... Can be combined using diversity combining techniques ... EE359 Discussion Session 9 OFDM,

EE359 Discussion Session 9OFDM, Spread Spectrum, Multiuser Systems

December 6, 2017

EE359 Discussion 9 December 6, 2017 1 / 1

Outline


Last discussion session

MIMO beamforming

Diversity multiplexing tradeoff

MIMO receiver design

Multicarrier modulation

This session

OFDM

Spread spectrum

Multiuser systems


Outline


Brief recap on multicarrier modulation (MCM)

Idea

Divide large bandwidth into smaller chunks and use narrowband signals

Pros and cons

Takes care of intersymbol interference (ISI)

Guard band versus spectral efficiency (SE)I No guard bands, high SE (e.g. OFDM): Overlapping subcarriers,

sensitive to timing/frequency offsetI Large guard bands, low SE (e.g. GSM FDM): Non overlapping

subcarriers, less stringent synchronization requirements

Multiplexing subcarriers

Analog: Use a separate modulator for each subcarrier and sum upsignals

Digital: Use a single modulator and then use a multiplexingarchitecture based on DFT (e.g. OFDM)


Homework 8

Problem 1

Repetition coding makes an error if more than half of the received bits arein error. Moreover the error event is dominated by error events in channelswith low SNRs.


Definition

DFT

Given N points x[0], . . . , x[N − 1], the DFT is given by

X[k] =

N−1∑i=0

x[i]e−j2πki/N =

N−1∑i=0

x[i]ωki, 0 ≤ k ≤ N − 1

Define Qω ∈ CN×N Qω,k,i = ωki 0 ≤ k, i < N , x ∈ CN s.t. xi = x[i],and X s.t. Xi = X[i]. If N is large, independent fading.

Some facts

X = Qωx

x = (Qω)−1X = Qω−1X =∑N−1

i=0 X[i]ω−ki (IDFT relation)

Matrix vector multiplication normally Θ(N2), but FFT/IFFTalgorithms allow Θ(N logN) complexity


FDM block diagram

Σ s(t)

Modulator 0 D/A s0(t)

cos(2πf0t)

Modulator . . . D/A s...(t)

cos(2πf...t)

Modulator N − 1 D/A sN−1(t)

cos(2πfN−1t)

Demodulator i LPF and A/D r(t)

cos(2πfit)

Figure: FDM system with N subchannels


OFDM block diagram

Modulator S/P IFFT CP, P/SCP, P/S D/A s(t)

cos(2πf0t)

X[n] x(t)

x0X0

x...X...

xN−1XN−1

Demodulator P/S FFT S/P, CP

removal

S/P, CP

removal

LPF and A/D r(t)

cos(2πf0t)

H[n]X[n] x(t)

x0H0X0

x...H...X...

xN−1HN−1XN−1

Figure: OFDM block diagram


Need for cyclic prefix

Effect of channel

Channel does linear convolution, i.e.,

y[n] = h[n] ∗ x[n] + ν[n]

Not multiplicative under DFT, which needs circular convolution

Rationale for cyclic prefix (CP)

Circular convolution can be simulated for finite support h[n] byadding CP to x[n]

Math: y[n] = h[n] ~ x[n] =∑N−1

m=0 h[m]x[n−m], where x[n] =x[n mod N ]

If |h[n]| = 0 for n /∈ [0, µ], CP of length µ suffices to “simulate”circular convolution/remove ISI


Final OFDM frame

X0 X1 X... XN−1

Frequency domain

xN−µ x... xN−1 x0 x... xN−1

Time domain (CP in red)

Some points

At the receiver simply discard the CP

One may also transmit null sequences instead of CP to “simulate”circular convolution (receiver processing different) (Problem 2,Homework 8)

Some features of OFDM based implementation

Efficient ISI removal ,

Inefficiency due to CP /


Equivalent matrix representation of OFDM

Effect of channel (linear convolution)

y = Hx + ν where H is Toeplitz, yi = y[i], xi = x[i]

Effect of equivalent channel (circular convolution)

y = Hx + ν where H is circulant

Fact about circulant matrices

They can be diagonalized by the DFT matrix Qω!

OFDM can be thought of as transmit precoding and receiver shapingusing Qω and QH

ω

Homework 8

Problem 3 explores this connection


OFDM Summary

Pros

Efficient ISI removal

Efficient implementation using FFT/IFFT

In a perfect system, subchannels are uncoupled

Cons

High peak to average power ratio

Sensitive to timing/frequency offset (which may couple adjacentsubchannels)


Outline


Introduction

Idea

Spread a narrowband signal over a wider band

Advantages

Signal not distinguishable from noise floor (military applications)

Narrowband interference rejection (again military applications)

Cellular resource allocation easier as compared to FDM/TDM

Disadvantages

Wideband receivers

Non orthogonality between spreading sequences (need equalization )

Two common incarnations

FHSS (Frequency hopping spread spectrum) (e.g. Bluetooth)

DSSS (Direct sequence spread spectrum) (e.g. CDMA in 3G )


DSSS

Idea

At transmitter: Use a spreading code (also known as chip sequence)sc(t) of bandwidth B = 1/Tc (sometimes called chip rate) with whichto multiply narrowband signal g(t) of duration Ts = 1/Bc

At receiver: Take the integral of r(t)sc(t) over time Ts

Desirable properties for sc(t)

Autocorrelation function ρ(τ) , 1/Ts∫ Ts0 sc(t)sc(t− τ)dt = δ(τ):

gives multipath rejection

Orthogonality∫ Ts0 sc1(t)sc2(t)dt = 0 for c1 6= c2: inter user

inteference rejection

Practical properties for sc(t)

Given finite Ts and finite B, cannot have many orthogonal sequences

Design spreading codes with approximate orthogonality


A specific sc(t)

Tc

Ts

1

-1

sc(t) in the time domain

1/Tc

Sc(f) frequency domain

At transmitter: Send s(t) = g(t)sc(t), g(t) narrowband

At receiver: Compute∫ Ts0 r(t)sc(t)dt


Narrowband interference rejection properties

Narrowband interference

Let g1(t) be narrowband, then system model is r(t) = s(t) + g1(t)

Spreading action at receiver

r(t)sc(t) = g(t) + g1(t)sc(t)

1/Tc

1/Ts

Figure: Frequency dom. of r(t)sc(t). Integration ≡ lowpass filtering

Narrowband interference rejection

Thus g1(t) is reduced by a factor of approximately Ts/Tc (spreading factor)


Homework 8

Problems 4

Explore the statistics of interference and noise in the spread spectrumsystem


Multipath (ISI) rejection properties

Multipath

Let h(t) = δ(t) + αδ(t− τ), then r(t) = s(t) + αs(t− τ)

Action at receiver

1/Ts∫ Ts0 r(t)sc(t)dt = gρ(0) + α/Tsρ(τ)

Autocorrelation function ρ(τ) of sc(t)

Smaller the support of ρ(τ), better is multipath/ISI rejection

Larger the support of ρ(τ), better is synchronization at receiver


Rake receivers

Idea

Using spreading codes with good ISI rejection, we can distinguish differentmultipath components!

Some facts

RAKE receiver simply gathers energy from different multipathcomponents with different delays

Different branches of the RAKE receiver synched to a different delaycomponent

Can be combined using diversity combining techniques (MRC/SC,etc.)


Homework 8

Problem 5

Use RAKE receivers with different diversity combining schemes

Problem 6

Sharing frequency, time or code resources for multi-user communications


Outline


Multiuser Communications

Centralized schemes - Resource divided in frequency, time, or codespace and allotted to users. (Cellular systems)

Decentralized schemes - Users attempt to communicate throughrandom access, if there are no collisions, it is successful. Else,transmit with random delay.

Cellular systems - Area divided into cells, different frequencies (reuse)could be assigned to them.


Homework 8

Problem 7

In centralized control, there are no collisions. For part (b), writing theindicator function that user i transmits in slot j would help - takingexpectations of these functions would throughput.

Problem 8

Frequency divided in half along adjacent cells. Interference is calculatedfrom closest cells.


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EE359 Discussion Session 9 OFDM, Spread Spectrum ... · Narrowband interference rejection ... Can be combined using diversity combining techniques ... EE359 Discussion Session 9 OFDM,